\begin{section}{Craighero-Gattazzo Surface over a Finite Field of Characteristic 5}

Let $\mathcal{F}$ denote the Craighero-Gattazzo quintic, and let $\mathcal{F}_{5}$ denote the Craighero-Gattazzo quintic after reduction modulo 5.

Note\footnote{For a more detailed view of how this was done, see our Macaualay2 script online at http://code.google.com/p/csboyd/source/browse/trunk/notebk/math.AG/reu/cgquintic/src/char5.m2} that $\mathcal{F}_{5}$ is obtained by making the substitutions 

$$a = r^2,$$
$$b = -\frac{2}{7}r^2 + \frac{13}{7}r + \frac{18}{7},$$
$$c = \frac{73}{49}r^2 + \frac{75}{49}r + \frac{92}{49},$$
$$e = -\frac{1}{7}r^2 + \frac{24}{7}r + \frac{9}{7},$$
$$f = \frac{181}{49}r^2 + \frac{241}{49}r + \frac{163}{49},$$
$$m = \frac{3}{7}r^2 + \frac{5}{7}r + \frac{1}{7}.$$

 We then multiply $\mathcal{F}$ (a polynomial over $\mathbb{Q}$) by 49 to cancel denominators (making it a polynomial over $\mathbb{Z}$) and setting $r = 8$ (since $(8^3 + 8^2 -1)(\textrm{mod} 5^2) \equiv 0$) and then reducing $\mathcal{F}$ modulo 5.

Let $T(n,f)$ denote the $n$th order Taylor expansion of the polynomial $f$, and let $T_{5}(n,f)$ denote the $n$th order Taylor expansion of $f$ after reduction modulo 5.

Take the plane quadric $$q = xy-xz-yz-xt+yt+zt.$$ 

Take the linear form $$l = x+t+z+t.$$ 

To get the $5$-adic expression $$F = q^{2}l + 5T_{5}(1,\mathcal{F}_{5}).$$

Let $V(I)$ be the ideal generated by the intersection of $\mathcal{F}_{5}$ and $q$. Using Macaulay2, we see that the decomposition of the singular locus of $V(I)$ gives the intersection of ideals 

$$\{(z,y,x), (t,z,y), (t,z,x), (t,y,x), (z + 2t, y + t, x - 2t), (z + t, y - t, x + t)\} \subset \mathbb{P}^{3}(\mathbb{Z}/5\mathbb{Z}).$$

Let $V(J)$ be the ideal generated by the intersection of $F$ and $q$. The decomposition of the singular locus of $V(J)$ gives the ideal

$$\{(y + t, x - 2t, z + 2t)\} \subset \mathbb{P}^{3}(\mathbb{Z}/5\mathbb{Z})$$

\end{section}
